# What is the Fourier transform of the product of two shifted-input functions?

I have the following expression

$$f(x+\delta)g(x+\delta)$$

I need to find the Fourier Transform. I know for

$$f(x+\delta)$$

the Shift Theorem states that the transform is $$e^{ik\delta}\hat{f}(k)$$, but I'm not sure how to apply the Theorem to the product above. The Fourier Transform of a product is a convolution, I know this.

My initial guess would be to apply the Shift Theorem to each term, then "bring them together" with a convolution giving

$$e^{ik\delta}\hat{f}(k)\star e^{ik\delta}\hat{g}(k)$$

That being said, it is a guess and I would like to find clarity.

If $$h(x)=f(x)g(x)\overbrace{\implies}^{FT}H(\omega)=\int_{-\infty}^\infty F(u)G(\omega-u)du$$then$$h(x+\delta)=f(x+\delta)g(x+\delta)\overbrace{\implies}^{FT}\hat{H}(\omega)=e^{j\delta \omega}\int_{-\infty}^\infty F(u)G(\omega-u)du$$